Distributions

Power

Difference Between Proportions

Statistics problems often involve comparisons between two
independent sample proportions. This lesson explains how to compute
probabilities associated with differences between proportions.

Difference Between Proportions: Theory

Suppose we have two
populations
with proportions equal to P1 and P2. Suppose
further that we take all possible
samples
of size n1 and n2. And finally, suppose that the
following assumptions are valid.

The size of each population is large relative to the sample
drawn from the population. That is, N1 is large relative
to n1, and N2 is large relative
to n2. (In this context, populations are considered to
be large if they are at least 20 times bigger than their sample.)

The samples from each population are big enough to justify using a
normal
distribution to model differences between proportions. The sample
sizes will be big enough when the following conditions are met:
n1P1> 10,
n1(1 -P1) > 10,
n2P2> 10, and
n2(1 - P2) > 10. (This criterion requires that at least 40 observations
be sampled from each population. When P1 or P2 is more extreme than 0.5, even
more observations are required.)

The samples are
independent;
that is, observations in population 1 are not affected by observations
in population 2, and vice versa.

Given these assumptions, we know the following.

The set of
differences between sample proportions will be normally
distributed. We know this from the
central limit theorem.

The
expected value of the difference between all
possible sample proportions
is equal to the difference between population proportions. Thus,
E(p1 - p2) = P1 - P2.

The standard deviation of the difference between sample
proportions (σd) is approximately equal to:

σd =
sqrt{ [P1(1 - P1) / n1] +
[P2(1 - P2) / n2] }

It is straightforward to derive the last bullet point, based on material
covered in previous lessons. The derivation starts with a recognition
that the variance of the difference between independent random variables is
equal to the sum of the individual variances. Thus,

σ2d =
σ2P1-P2 =
σ21 + σ22

If the populations N1 and N2 are both large
relative to n1 and n2, respectively,
then

Difference Between Proportions: Sample Problem

In this section, we work through a sample problem to show how to apply
the theory presented above. In this example,
we will use Stat Trek's
Normal Distribution Calculator
to compute probabilities.

Normal Distribution Calculator

The normal calculator solves common statistical problems, based on the normal
distribution. The calculator computes cumulative probabilities, based on three
simple inputs. Simple instructions guide you to an accurate solution, quickly
and easily. If anything is unclear, frequently-asked questions and sample
problems provide straightforward explanations. The
calculator is free. It can found in the Stat Trek
main menu under the Stat Tools tab. Or you can tap the button below.

In one state, 52% of the voters are Republicans, and 48% are Democrats.
In a second state, 47% of the voters are Republicans, and 53% are
Democrats. Suppose 100 voters are surveyed from each state.
Assume the survey uses simple random sampling.

What is the probability that the survey
will show a greater percentage of Republican voters in the
second state than in the first state?

(A) 0.04
(B) 0.05
(C) 0.24
(D) 0.71
(E) 0.76

Solution

The correct answer is C. For this analysis, let P1 =
the proportion of Republican voters in the first state,
P2 = the proportion of Republican voters in the second state,
p1 = the proportion of Republican voters in the
sample from the first state, and
p2 = the proportion of Republican voters in the
sample from the second state. The number of voters sampled from
the first state (n1) = 100, and the number of voters
sampled from the second state (n2) = 100.

Find the probability. This problem requires us to find the
probability that p1 is less than p2.
This is equivalent to finding the probability that
p1 - p2 is less than zero. To find this
probability, we need to transform the random variable
(p1 - p2) into a
z-score.
That transformation appears below.

Therefore, the probability that the survey
will show a greater percentage of Republican voters in the
second state than in the first state is 0.24.

Note: Some analysts might have used the t-distribution to compute probabilities
for this problem. We chose the normal distribution because the population variance was known
and the sample size was large. But it would not have been wrong to use the t-distribution. In a previous lesson, we offered some guidelines for
choosing between the normal and the t-distribution.